A velocity vector is first and foremost a vector. That is, it has both a magnitude and a direction. I will not discuss vectors here; that will be left up to the vector node. Suffice it to say that, in Cartesian coordinates, a velocity vector can be written as a set of components. A 3-dimensional velocity vector, v can be written in the form of <vx, vy, vz>.
The velocity vector is a vector which represents the velocity of an object. The velocity of an object is simply the distance travelled by the object per unit time. As stated, this is only a magnitude, but velocity is in reality directional in nature. More precisely stated, a velocity is the time derivative of the positional function of the object. That is, it is the change in the position of the object per unit time. Thus, if supplied a function which gives the position, r of an object at any given point in time, say <rx, ry, rz>, the velocity can be determined by:
- vx = d/dt(rx)
- vy = d/dt(ry)
- vz = d/dt(rz)
More
generally stated,
vi = d/dt(ri), where ri and vi are the
magnitude of
r and
v vectors,
respectively, in the
direction the i-
axis. For example, for vx and rx, i = 1. For vy and ry, i = 2. Because of the
componentwise
nature of the
vector derivative, the above can be written more simply as:
v = d/dt(
r).
Naturally, then, r can be determined by integrating each component of v. The acceleration vector, a is just the time derivative of v, and thus v can be found by integrating v. A quick overview:
I | acceleration /\
N | |
T | | D
E | | E
G | velocity | R
R | | I
A | | V
T | | E
E \/ position |
Although not
necessarily apparent,
velocity is a
vector, and thus, a
velocity vector is simply a
vector which
stores a
velocity. The
velocity is not in the
vector itself; it's in the
use or
interpretation of the
vector.